In the following sections we demonstrate the importance of constructing neural ODEs with a linear and a nonlinear term on the VBE and the KSE. We examine the VBE to highlight the ability of this method to accurately capture the shock behavior seen at low viscosity and to accurately capture the energy spectrum. Then, we consider the KSE to show how our approach is needed to keep the long-time trajectories of a chaotic dynamical system on the attractor. This then leads to a demonstration of reduced order modeling via Galerkin and nonlinear Galerkin using the eigenvectors of the linear term we discover. In all cases results are shown for test data that was not used for training any of the models. Neural ODEs are trained using PyTorch [1]} with code modified from Chen et al. [2]}.
